The. Consortium. Continuum Mechanics. Original notes by Professor Mike Gunn, South Bank University, London, UK Produced by the CRISP Consortium Ltd

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1 The C R I S P Consortium Continuum Mechanics Original notes b Professor Mike Gunn, South Bank Universit, London, UK Produced b the CRISP Consortium Ltd

2 THOR OF STRSSS In a three dimensional loaded bod, there are si indeendent comonents of stress at a oint Shear stress notation: ab face of cube direction B moment euilibrium:,, In an stressed material there are alwas three mutuall erendicular lanes on which the shear stresses are ero. These are called the rincial lanes. The direct stresses acting on these lanes are called rincial stresses (,, ). c In a stressed bod the comonents of stress are,,,, and. We can also describe the stress state with resect to another set of aes (e.g. a, b, c). b a

3 Stress invariants are functions of the stress comonents which are indeendent of the ais sstem chose. For eamle, c b a is a stress invariant. Another stress invariant is where: ( ) ( ) ( ) ( ) ( ) ( ) ca bc ab a c c b b a Note that these general definitions of and reduce to those given later for triaial stress conditions (see lecture on Cam-cla). We will encounter these invariants more freuentl eressed in terms of the rincial stresses, i.e. ( ) ( ) ( ) Note,, are also stress invariants. There is a simle geometric interretation of and in rincial stress sace. The line is called the sace diagonal or the hdrostatic ais OA ( ) AB AB B A O (,, )

4 LASTICIT Generalised Hooke s Law: ε ν ν ε ν ν ε ν ν γ γ γ ( ν) ( ν ) ( ν ) PLASTICIT INTRODUCTION The theor of elasticit allows the calculation of stresses and strains in a loaded bod when the bod is linear and elastic. The theor of lasticit allows the calculation of stresses and strains in a loaded bod when lastic ielding takes lace. Histor (lasticit) 8 Cauch invents concets of stress and strain 885 Boussines gives his solutions for elastic stress distributions (still used in geotechnical engineering toda) (For 00 ears mathematicians slave awa obtaining elasticit solutions...) 956 Finite element method invented (mathematicians are now redundant) 4

5 Histor (Plasticit) 77 Coulomb identifies two comonents in the strength of soil - cohesion and friction 864 Criteria are ut forward for (Tresca) limits to the elastic behaviour of metals 9 (Von Mises) 940 s tes of statement are now seen to be necessar to comletel describe lastic stress-strain relations: a) ield criteria b) flow rule c) hardening law 950 R. Hill s book Mathematical Theor of Plasticit is ublished. PLASTIC BHAVIOUR FOR ON- DIMNSIONAL LOADING Direct strain in direction is ε. B D Tical stress -strain relation for an elastic, work hardening lastic material (e.g. metal allo) A O C OA is elastic. A is a ield oint. is the uniaial ield stress. BC and D are elastic unloading and reloading (arallel to OA). On reloading to B the ield stress has increased to. The material is harder and has strain (or work) hardened. 5

6 In describing lastic behaviour, the following simlifications ( idealisations ) are often made. lastic, erfectl lastic (steel behaviour is uite like this) lastic, linear strain hardening lastic Rigid-lastic (often assumed in collase calculations) ILD FUNCTIONS - BASIC IDAS is held constant and is increased until ielding starts (or vice versa) The combinations of and that cause ielding are described b a ield function. ield functions can be reresented b lines in D stress sace and surfaces in D stress sace. 6

7 For elastic-erfectl lastic behaviour: LASTIC Plastic ielding on this ield curve or surface Stress states inside the ield surface are elastic. Stress states outside the ield surface are imossible, b definition. If strain hardening takes lace, there are two ossibilities: A B A B C O O C The ield surface eands uniforml - this is called isotroic hardening A B A B C ' O O C' The ield surface is dragged along - this is called kinematic hardening. The differences between these two assumtions become imortant if unloading takes lace. 7

8 ILD FUNCTIONS - XAMPLS a) TRSCA ielding takes lace when the maimum value of -, -, - euals a critical value (k). This can be interreted (b considering Mohr s circles) as being euivalent to limiting the maimum shear stress on an lane in the material to being less than or eual to k. k k Consider the ield condition in (, ) stress sace when 0. k k k k k For triaial stress conditions and we can lot the ield function in (, ) sace. k k k 8

9 b) VON MISS Plastic ielding takes lace when: ( ) ( ) ( ) where is the ield stress in uniaial tension. Note (from our earlier definition of ) that the above euation can be written as: Structural and mechanical engineers call the stress arameter either: i) the Von Mises stress, ii) the Von Mises euivalent stress, or iii) the euivalent stress and use the smbol rather than. Von Mises ield function in (, ) sace when 0 : This is an ellise with its major ais along. The Tresca ield criterion is shown thus for comarison. Clearl k. In triaial stress conditions and the ield function in (, ) sace is as shown: STRAINS For ever stress comonent or invariant there is a corresonding strain comonent so that the work done er unit volume in elastic deformation is / stress strain. W.D ε 9

10 Stress Strain ε ε ε γ γ γ ε ε ε Stress invariant Strain invariant (defined reviousl) ε (volumetric strain) ε ε ε ε ε ε (defined reviousl) ε (deviatoric strain) ε ( ε ε ) ( ε ε ) ( ε ε ) ( γ γ γ ) ε ( ε ε ) ( ε ε ) ( ε ε ) 0

11 PLASTIC STRAINS e The lastic strain is the strain which remains on comletel unloading the alied stress. ε ε e ε T ε lastic strain ε e elastic strain ε T total strain CALCULATION OF LASTIC STRAINS Suose that the soil is in some stress state ( [,, ] T ). An increment of stress is now alied to the soil ( ). The resulting strains, if the behaviour is elastic, can be calculated as ε C where C is a suare matri containing elastic constants. CALCULATION OF PLASTIC STRAINS In contrast to elastic behaviour, it is found that lastic strains are strongl deendent on the current total stresses. C Thus the lastic strains generated a ield oint C are not deendent on the direction of the stress ath when ielding starts. A X O B X i.e. aths OAC and OBC lead to the same lastic strains.

12 PLASTIC POTNTIALS Mathematicall, the te of behaviour described above can convenientl be eressed in terms of a otential function such that the derivatives of the otential function define the ratios of the lastic strains. Plastic strain comonents are often lotted in stress sace:, Current stress state often called the strain increment vector (, ), Mathematicall we write:, constant otential G d ε m () where G is the lastic otential, For metals the lastic otential function is the same as the ield function i.e. G F and hence we can write: F m d ε () (m is a scalar number which deends on the hardening law and the details of a articular analsis. It is called the lastic multilier). when G F we sa there is normalit (the strain increment vector is normal to the ield surface) - also called associated flow. uations like () and () above are called flow rules. The govern the ratios of the lastic strain comonents.

13 Flow Rule - amle with Von Mises (a) in, sace when 0 F (, ) 0 dε dε F m m F m m ( ) ( ), dε X X, dε B insection the strain increment vector is erendicular to the ellise reviousl described. (b) in, sace, dε ( ) ielding takes lace with ero volumetric strains dε 0 lastic P, dε

14 () The Von Mises ield Surface c Hdrostatic ais b a () The Tresca ield Surface c Hdrostatic ais b a 4

15 () The Drucker-Prager ield Surface c Hdrostatic ais b a (4) The Mohr-Coulomb ield Surface c Hdrostatic ais b a 5